SINGLY REINFORCED SECTION :
When the beam us loaded it can suffer through two different type of beam(mode of failure):
1.Ductile Failure &
2.Brittle Failure
Now, in case of a brittle failure of beam when a beam gets over-stressed, then till the point of over-stressing any major cracks aren’t developed in beam, but as soon the point is passed, suddenly the beam fails.Well thus doesn’t seem good h..!! Hence we always design the beam to fail in the ductile fashion, consequently giving us more time to notice that the beam is over-stressed and it’s the time that we should do some repairs or evacuate the building.
Now, how do we make sure that the beam will fail in ductile manner. This can only possible when the strength of the steel will be less than compression strength of the concrete. This means that when the ultimate loads are reached, as the total strength of the steel will be pretty less than the strength of concrete, steel will try to strain more than what the concrete will experience strain in compression. Now we all know that steel is ductile and able to handle higher strains without any kind of failure. So this is how we can get a ductile response of beam. In this case, Concrete is all in under control while steel is straining itself.
But suppose if there are certain restrictions like we cannot use a section greater than say 25″ x 25″ & we have to deal with very high moments. Now this high moment will tend to increase the tension steel demand & consequently a bigger concrete block will be required in compression resulting in pushing the neutral axis further down. Now this is an important part. One might think that so what if neutral axis go further down, I have the whole concrete beam which can take the compression. But wait..!!
As a neutral axis starts shifting down, the strains in the extreme fiber starts to increase. And this will be our concern in case of the compression fiber. Now concrete is brittle, so at start/first it will not show that much impact. But this high value of the strain will cause concrete to crush itself which will lead to the ultimate failure and this will be sudden. That’s why we can’t put more steel than a certain amount.
So, what decides this limit of steel ? Well, it’s all experimental based approach & to an extent we can prove it mathematically too. Balancing tension & compressive forces, take the depth of a compression block, now draw the strains and look if the strain in the concrete is more than the allowable strain. But in case of designing by us, then some codes specify the maximum r/f ratio in beam whereas some codes specify the maximum allowable neutral axis depth that concrete beam can achieve. And if the desing is under this, thenductile failure in a beam will occur and we are safe.
DOUBLY REINFORCED SECTION :
Now as we mentioned before that we are restricted to a 24″ x 24″ section & we have pretty heavy moment & we are exceeding the limits mentioned in the code. So now the only option is to provide thr compression reinforcement & add some tension steel. Now the extent to which we are adding the compression reinforcement will tell us whether the beam is brittle or ductile.
Suppose the limit for tension reinforcement for singly reinforced beam is X, & to resist this moment we have to add extra 0.5X of tension reinforcements. We also decide to include 0.8X of compression reinforcement. Hence, now we have 1.5X(1X+.5X) of tension reinforcement & 0.8X of compression reinforcement provided along with compression stress block. So now what happens is that this 0.8X of compression r/f will tend to balance the effect of 0.5X of tension r/f. As we have added 0.8X of compression r/f, the beam doesn’t experience a very high compression strain compared to tension strain. Thus the compression bars will be under lower stress than its yield point. Thus to balance a fully stressed 0.5X of tension r/f we will need a higher amount of compression r/f . Now, the concrete block is responsible to resist the X amount of the tension r/f which is a limit for ductile behavior. Hence in this case the doubly reinforced beam will act as a ductile beam and so we can say that it is an under-reinforced beam/section. All good..!!
But suppose we decide to add only 0.4X of extra compression r/f. Well This will counter max of 0.4X of tension r/f in worst case scenario(it may be lower). So now, the plain concrete stress block will be responsible to resist 1.1X [1X + (.5X-.4X) ]of tension r/f and this is above the limit resulting in brittle failure. So in this case, even though the beam is doubly reinforced, it will experience a brittle failure i.e. it’s an over-reinforced section.
